s13 - Math 5615H: Introduction to Analysis I. Fall 2011...

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Unformatted text preview: Math 5615H: Introduction to Analysis I. Fall 2011 Homework #13. Problems and Solutions. #1. If f is a continuous mapping of a metric space X into a metric space Y , prove that f ( E ) f ( E ) for every set E X . ( E denotes the closure of E ). Show, by an example, that f ( E ) can be a proper subset of f ( E ). Proof. If p E , then f ( p ) f ( E ) f ( E ). If p E \ E , then p is a limit point of E , i.e. there is a sequence { p n } E converging to p . By continuity of f , we have f ( p n ) f ( p ). Since f ( p n ) f ( E ) for each n , this implies f ( p ) f ( E ). In any case, we have f ( p ) f ( E ) for each p E , i.e. f ( E ) f ( E ). The above inclusion can be proper. For example, the function f ( x ) := 1 /x is continuous on E = E = [1 , ), while f ( E ) = (0 , 1] f ( E ) = [0 , 1] , f ( E ) = f ( E ). Note that the proper inclusion f ( E ) f ( E ) is impossible if f is continuous on a compact set E (why?) #2. Show that every increasing convex function of a convex function is convex....
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s13 - Math 5615H: Introduction to Analysis I. Fall 2011...

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